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Full text of "Treatise On Analysis Vol-Ii"

8   CONVOLUTION OF A MEASURE AND A FUNCTION       279

6. (a) Let /x be a measure on a locally compact group G. Show that JLX * v = v * /x for all
measures v such that both jit, * v and v * jit are defined, if and only if ft * £x = £x * /x
for all x e G.

(b)   Suppose that G is compact. If JLI is any measure on G, show that the measure
/x^ defined by

(where j3 is a Haar measure on G) satisfies /^ *v = v*/zfcl for every measure v on G.

7.    Let G be a compact group and j8 the Haar measure on G for which /3(G) = 1. Let /z
be a positive measure on G such that /z(G) — 1 and /x > c/3, where 0 < c < 1 . Show that
llj"-*B — /3|l ;S 2M(1 — c)", where jit*" denotes the convolution product of n measures
equal to p (use (14.6.3)).

8.    Let r be a real number such that 0 < r < J. For each integer « > 1, let pn,r denote the
measure J(£rn + e_rfl) on R, and let jun, r denote pi, r * pa. r * " " * p«, r •

(a)    Show that the sequence (^n, r)n>i converges vaguely to a measure /A on R with
support contained in I = [—1, 1] (prove that for every interval U c R the sequence
(jLtn,r(U)) converges).

(b)    Show that, if r < J, the measure /x is disjoint from Lebesgue measure on R, but
that /xi/2 is the measure induced on I by Lebesgue measure.

(c)    Let i/1/4 be the image of jLt1/4 under the homothety t\-*2t on R. Show that
j^-i/4 * vi/4 = ju.j/2, although /xl/4. and vi/4 are each disjoint from Lebesgue measure
(use Problem 4(b)).

8. CONVOLUTION OF A MEASURE AND A FUNCTION

In the rest of this chapter we shall fix once and for all a left Haar measure /?
on the group G. If/is any mapping of G into R or C, the norm Np(/)
(p ~ 1, 2 or +00) is taken with respect to the measure /?.

(14.8.1) Let PL be a measure on G and let f be a (complex) locally f$-integrable
function on G (13.13.1). For the measures ^ and f- ft to be convolvable, it is
necessary and sufficient that there should exist a ^-negligible set N such that the
function s\-+f(s~~ lx) is fi-integrable for all x $ N, and the function

H"1

(defined almost everywhere with respect to /?) is locally $-integrable. When this
condition is satisfied, the function g(x) =*[f(s~1x) dn(s), defined almost every-
where with respect to /?, is locally p-integrable, and in * (/• /?) is equal to g • ft.

We shall first prove the following lemma:* vn ->• ft * v (use Problem 2 of Section 13.20, and EgorofTs theorem).